The connection formula of the \(q\) -hypergeometric series is well known. See the equation (4.5.2) on p. 121 in [1]}
for example. From Proposition REF , the connection formula of
the degenerate 4-point \(q\) -conformal block reads as
\(\mathcal {F}\left({\frac{1}{N}\phantom{0000}\theta _1\atop \sigma _2\phantom{00}\sigma _2+h_i\phantom{00}\sigma _0};\,x_1,x_2\right)\nonumber \\ \qquad {}=\sum _{j=1}^N\mathcal {F}\left({\theta _1\phantom{0000}\frac{1}{N}\atop \sigma _2\phantom{00}\sigma _0-h_j\phantom{00}\sigma _0};\,x_2,x_1\right) {\mathcal {B}}_{j,i}\left[\begin{matrix}\theta _1&\frac{1}{N}\\\sigma _2&\sigma _0\end{matrix}\Bigl | \frac{x_2}{x_1}\right]q^{N\theta _1^2/2-\theta _1/2}\bigg (\frac{x_2}{x_1}\bigg )^{\theta _1},\) 
